Lesson Notes By Weeks and Term v3 - Junior Secondary 2
Approximation
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Approximation
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Subject: General Mathematics
Class: Junior Secondary 2
Term: 2nd Term
Week: 2
Theme: Basic Operations
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
Approximate numbers to any given degree of accuracy Solve quantitative reasoning problems related to approximation of numbers.
2.1 Definition of Approximation: Approximation is the process of finding a number that is close enough to an exact value for a particular purpose. It involves rounding off numbers to a specified degree of accuracy, typically by dropping digits after a certain point and adjusting the last retained digit based on the value of the first dropped digit. 2.2 General Rules for Rounding Off Numbers: When rounding a number, observe the digit immediately to the right of the digit to which the number is to be rounded. If this digit is 5 or greater (5, 6, 7, 8, 9), add 1 to the digit to be rounded, and drop all subsequent digits. If this digit is less than 5 (0, 1, 2, 3, 4), keep the digit to be rounded as it is, and drop all subsequent digits. 2.3 Approximation to Decimal Places (d.p.): Decimal places refer to the number of digits after the decimal point. To approximate a number to a specified number of decimal places:
1. Identify the digit at the specified decimal place.
2. Look at the digit immediately to its right.
3. Apply the general rounding rules: If the digit to the right is 5 or more, add 1 to the digit at the specified decimal place and drop all digits to its right. If the digit to the right is less than 5, keep the digit at the specified decimal place as it is and drop all digits to its right. Worked
Examples: Example 1: Approximate ₦456.7839 to 2 decimal places (2 d.p.).
Step 1: Identify the digit at the second decimal place. This is '8'.
Step 2: Look at the digit immediately to its right. This is '3'.
Step 3: Since '3' is less than 5, the digit '8' remains unchanged. Drop all digits after '8'.
Solution: ₦456.78 Example 2: Round 2.3456 to 3 decimal places (3 d.p.).
Step 1: Identify the digit at the third decimal place. This is '5'.
Step 2: Look at the digit immediately to its right. This is '6'.
Step 3: Since '6' is 5 or greater, add 1 to '5', making it '6'. Drop all digits after the third decimal place.
Solution: 2.346 Example 3: A tailor used 3.875 metres of Ankara fabric. Approximate this length to the nearest tenth of a metre (1 d.p.).
Step 1: The nearest tenth of a metre means 1 decimal place. The digit at the first decimal place is '8'.
Step 2: The digit to its right is '7'.
Step 3: Since '7' is 5 or greater, add 1 to '8', making it '9'. Drop all digits after '9'.
Solution: 3.9 metres 2.4 Approximation to Significant Figures (s.f.): Significant figures are the digits in a number that carry meaning or contribute to its precision.
Rules for identifying significant figures: Non-zero digits: All non-zero digits are significant. (e.g., 123 has 3 s.f.)
Zeros between non-zero digits: Zeros between non-zero digits are significant. (e.g., 102 has 3 s.f.)
Leading zeros: Zeros before non-zero digits (leading zeros) are NOT significant. They only indicate the position of the decimal point. (e.g., 0.0012 has 2 s.f.; 0.23 has 2 s.f.)
Trailing zeros: After a decimal point: Trailing zeros AFTER a decimal point are significant. (e.g., 1.200 has 4 s.f.; 0.50 has 2 s.f.)
Without a decimal point: Trailing zeros in a whole number without a decimal point are generally NOT significant unless specified by context (e.g., 1200 could have 2, 3, or 4 s.f. depending on context. To specify 3 s.f., it might be written as 1.20 x 10^3). For JSS2 level, assume trailing zeros in whole numbers without decimal points are generally not significant unless a decimal point is explicitly placed (e.g., 1200. has 4 s.f.). A safer approach for whole numbers is to assume non-significant unless scientific notation is used. To approximate a number to a specified number of significant figures:
1. Identify the significant figures starting from the first non-zero digit.
2. Locate the digit at the specified significant figure position.
3. Look at on context. To specify 3 s.f., it might be written as 1.20 x 10^3). For JSS2 level, assume trailing zeros in whole numbers without decimal points are generally not significant unless a decimal point is explicitly placed (e.g., 1200. has 4 s.f.). A safer approach for whole numbers is to assume non-significant unless scientific notation is used. To approximate a number to a specified number of significant figures:
1. Identify the significant figures starting from the first non-zero digit.
2. Locate the digit at the specified significant figure position.
3. Look at the digit immediately to its right.
4. Apply the general rounding rules: If the digit to the right is 5 or more, add 1 to the digit at the specified significant figure position. If the digit to the right is less than 5, keep the digit at the specified significant figure position as it is.
5. Replace any subsequent digits before the decimal point with zeros to maintain place value. Drop any subsequent digits after the decimal point. Worked
Examples: Example 4: Approximate 34,567 to 3 significant figures (3 s.f.).
Step 1: Identify the first three significant figures: '3', '4', '5'. The third significant figure is '5'.
Step 2: Look at the digit immediately to its right. This is '6'.
Step 3: Since '6' is 5 or greater, add 1 to '5', making it '6'.
Step 4: Replace the remaining digits ('6', '7') with zeros to maintain place value.
Solution: 34,600 Example 5: Round 0.007891 to 2 significant figures (2 s.f.).
Step 1: The first non-zero digit is '7'. The first two significant figures are '7', '8'. The second significant figure is '8'.
Step 2: Look at the digit immediately to its right. This is '9'.
Step 3: Since '9' is 5 or greater, add 1 to '8', making it '9'.
Step 4: Drop all digits after '9'. Leading zeros remain.
Solution: 0.0079 Example 6: The population of a village is 12,345 people. Approximate this to 1 significant figure (1 s.f.).
Step 1: The first significant figure is '1'.
Step 2: The digit to its right is '2'.
Step 3: Since '2' is less than 5, '1' remains unchanged.
Step 4: Replace the remaining digits ('2', '3', '4', '5') with zeros to maintain place value.
Solution: 10,000 people 2.5 Quantitative Reasoning Related to Approximation: Quantitative reasoning problems involving approximation require students to apply the concepts of decimal places and significant figures to solve practical scenarios, often presented as word problems or number series with a logical pattern requiring estimation. This assesses the ability to interpret, apply, and evaluate numerical information. Worked
Example: Example 7: A trader bought 5 bags of rice, each weighing 49.7 kg. If he wants to estimate the total weight to the nearest 10 kg, how would he do it?
Step 1: First, approximate the weight of one bag to the nearest whole number or 1 s.f. for easier calculation. 49.7 kg to nearest whole number: 50 kg (since 7 > 5, round 9 up, carries over to 4) 49.7 kg to 1 s.f.: 50 kg (first s.f. is 4, next is 9. 9 >= 5, so 4 becomes
5. Other digits become 0)
Step 2: Calculate the total estimated weight. Estimated total weight = 5 bags 50 kg/bag = 250 kg Step 3: Present the answer to the nearest 10 kg as required (which 250 already is). * Solution: The total estimated weight is 250 kg. 3.1 Introduction (10 minutes)
Teacher Activity: Begin by asking students about situations where they might not need an exact number, but rather an estimate (e.g., estimating the number of attendees at a wedding, the cost of items in a market, the time needed to travel). Introduce the term "approximation" as making a sensible guess or rounding a number.
Student Activity: Students share their experiences with estimation, engage in a brief discussion on why approximation is useful in everyday Nigerian life. 3.2 Explanation and Demonstration (20 minutes)
Teacher Activity: Explain the general rules for rounding off numbers, using a number line if necessary to visually represent "closer to". Introduce and define "decimal places" (d.p.). Demonstrate how to round numbers to 1, 2, and 3 decimal places using clear, step-by-step examples (e.g., fuel prices, exchange rates). Introduce and define "significant figures" (s.f.). Explain the rules for identifying significant figures, including zeros. Demonstrate how to round numbers to 1, 2, and 3 significant figures, using examples like population figures, distances. Emphasize the difference between d.p. and s.f. and when to use each. Introduce basic quantitative reasoning problems that require approximation.
Student Activity: Students actively listen, take notes, and ask clarifying questions. Copy down the rules and worked examples from the board. Practice identifying decimal places and significant figures in given numbers. 3.3 Guided Practice (15 minutes)
Teacher Activity: Provide a few practice problems for students to attempt in class. Circulate to monitor progress, identify common errors, and provide immediate feedback.
Example: "Approximate 7.145 to 2 d.p." "Approximate 0.0567 to 1 s.f." "A farmer harvests 2348 tubers of yam. Approximate this to the nearest hundred." Student Activity: Students work individually or in pairs to solve the practice problems. Share their answers and methods with the class. Correct their work based on teacher feedback. 3.4 Quantitative Reasoning Application (10 minutes)
Teacher Activity: Present a simple quantitative reasoning problem related to approximation. Guide students through the steps to solve it.
Example: "A vulcanizer charges ₦150.50 per tyre. If a customer needs 4 tyres pumped, approximately how much should the customer pay to the nearest N10?" (Teacher guides to round ₦150.50 to ₦151 or ₦150 first, then multiply by 4, then round the total).
Student Activity: Students attempt the problem, discuss strategies in small groups, and present their estimated solutions. 3.5 Conclusion and Recap (5 minutes)
Teacher Activity: Summarize the key concepts covered: rounding rules, decimal places, significant figures, and their application in quantitative reasoning. Emphasize the importance of approximation in real-life situations in Nigeria.
Student Activity: Students briefly recap what they have learned, highlighting the main differences between approximating to decimal places and significant figures.
Question 1: A petrol station sells fuel at ₦168.35 per litre. A customer bought 15 litres. i. Approximate the price per litre to the nearest kobo (2 d.p.). ii. Approximate the total cost to the nearest whole Naira.
Solution 1: i. Approximate the price per litre to the nearest kobo (2 d.p.). The price is ₦168.
3
5. We need to round to 2 decimal places. The digit at the second decimal place is '5'. The digit to its right is nothing (or 0 implied), so we consider '5' itself. According to rounding rules, if the digit at the specified place is 5 and there are no further digits, or the next digit is 0, it means it is exact. If rounding to the exact digit, and the problem asks for rounding, it can sometimes imply going further.
However, here "nearest kobo" is 2 d.p., and the number is already expressed to 2 d.p. If it were ₦168.354, then it would be ₦168.
3
5. If it were ₦168.348, it would be ₦168.
3
5. Since it's exactly ₦168.35, it remains ₦168.
3
5. Answer: ₦168.35 ii. Approximate the total cost to the nearest whole Naira. First, calculate the exact total cost: ₦168.35 15 = ₦2525.25 Now, approximate ₦2525.25 to the nearest whole Naira (0 d.p.). The digit at the first decimal place is '2'. Since '2' is less than 5, the whole number part '2525' remains unchanged.
Answer: ₦2525
Commentary: Part (i) highlights that if a number is already at the specified decimal place, it remains unchanged. Part (ii) shows a practical application of approximation after an initial calculation.
Question 2: The Federal Government budgeted ₦13,579,210,000 for a rural electrification project in Kwara State. i. Approximate this budget to 3 significant figures. ii. Approximate this budget to 1 significant figure.
Solution 2: i. Approximate this budget to 3 significant figures. The number is 13,579,210,
0
0
0. Identify the first three significant figures: '1', '3', '5'. The third significant figure is '5'. The digit immediately to its right is '7'. Since '7' is 5 or greater, add 1 to '5', making it '6'. Replace all subsequent digits with zeros to maintain place value.
Answer: ₦13,600,000,000 ii. Approximate this budget to 1 significant figure. The number is 13,579,210,
0
0
0. Identify the first significant figure: '1'. The digit immediately to its right is '3'. Since '3' is less than 5, '1' remains unchanged. Replace all subsequent digits with zeros to maintain place value.
Answer: ₦10,000,000,000
Commentary: This example demonstrates how large numbers, typical in government budgets, are approximated for easier communication and understanding at different levels of precision.
Question 3: A student measures the length of a classroom as 8.462 meters. i. Express this length to 1 decimal place. ii. Express this length to 2 significant figures.
Solution 3: i. Express this length to 1 decimal place. The number is 8.
4
6
2. The digit at the first decimal place is '4'. The digit to its right is '6'. Since '6' is 5 or greater, add 1 to '4', making it '5'. Drop subsequent digits.
Answer: 8.5 meters ii. Express this length to 2 significant figures. The number is 8.
4
6
2. The first two significant figures are '8', '4'. The second significant figure is '4'. The digit to its right is '6'. Since '6' is 5 or greater, add 1 to '4', making it '5'. Drop subsequent digits.
Answer: 8.5 meters
Commentary: This question highlights that sometimes approximation to a certain number of decimal places can yield the same result as approximation to a certain number of significant figures, depending on the number itself.
Question 4: Mama Bimpe buys 3.75 kg of yam, 2.15 kg of garri, and 1.8 kg of beans from the market. She wants to estimate the total weight of her groceries to the nearest kilogram.
Step 1: Approximate each item's weight to the nearest whole number (or 0 d.p.). * Yam: 3.75 kg. The digit at 1 d.p. is '7'. Since '7' >= 5, round 3 up to
4. So, 4 kg.
Example 1: Approximate ₦456.7839 to 2 decimal places (2 d.p.).
Step 1: Identify the digit at the second decimal place. This is '8'.
Step 2: Look at the digit immediately to its right. This is '3'.
Step 3: Since '3' is less than 5, the digit '8' remains unchanged. Drop all digits after '8'.
Solution: ₦456.78
Example 2: Round 2.3456 to 3 decimal places (3 d.p.).
Step 1: Identify the digit at the third decimal place. This is '5'.
Step 2: Look at the digit immediately to its right. This is '6'.
Step 3: Since '6' is 5 or greater, add 1 to '5', making it '6'. Drop all digits after the third decimal place.
Solution: 2.346
Example 3: A tailor used 3.875 metres of Ankara fabric. Approximate this length to the nearest tenth of a metre (1 d.p.).
Step 1: The nearest tenth of a metre means 1 decimal place. The digit at the first decimal place is '8'.
Step 2: The digit to its right is '7'.
Step 3: Since '7' is 5 or greater, add 1 to '8', making it '9'. Drop all digits after '9'.
Solution: 3.9 metres
2.4 Approximation to Significant Figures (s.f.):
Significant figures are the digits in a number that carry meaning or contribute to its precision.
Rules for identifying significant figures:
Non-zero digits: All non-zero digits are significant. (e.g., 123 has 3 s.f.)
Zeros between non-zero digits: Zeros between non-zero digits are significant. (e.g., 102 has 3 s.f.)
Leading zeros: Zeros before non-zero digits (leading zeros) are NOT significant. They only indicate the position of the decimal point. (e.g., 0.0012 has 2 s.f.; 0.23 has 2 s.f.)
Trailing zeros:
After a decimal point: Trailing zeros AFTER a decimal point are significant. (e.g., 1.200 has 4 s.f.; 0.50 has 2 s.f.)
Without a decimal point: Trailing zeros in a whole number without a decimal point are generally NOT significant unless specified by context (e.g., 1200 could have 2, 3, or 4 s.f. depending on context. To specify 3 s.f., it might be written as 1.20 x 10^3). For JSS2 level, assume trailing zeros in whole numbers without decimal points are generally not significant unless a decimal point is explicitly placed (e.g., 1200. has 4 s.f.). A safer approach for whole numbers is to assume non-significant unless scientific notation is used.
Market Transactions and Budgeting (Community/Economy): Nigerians frequently use approximation when buying goods at local markets. For example, when buying items like yams, garri, or vegetables, prices might be in fractions (e.g., "₦250 for three tubers and one small one"). Approximating allows buyers and sellers to quickly estimate total costs or change without complex calculations. For family budgeting, monthly expenses are often rounded to the nearest Naira or hundred Naira for easier tracking and planning. Construction and Engineering (Economy/Community): In construction, measurements for materials like cement, sand, gravel, and wood are often approximated. An engineer or builder might estimate that "about 10 bags of cement" are needed for a small project, or a length of wood is "roughly 3.5 meters." While precision is important, initial estimations use approximation to quickly assess project scope and material costs before exact measurements are taken. Reporting Statistics and Demographics (Community/Environment): Government agencies and news media in Nigeria frequently use approximation when reporting large figures such as population counts, election results, or national budget allocations. For instance, the population of a state might be given as "approximately 5 million" instead of 4,987,
6
5
4. This makes complex data more digestible and understandable for the general public and policymakers, allowing for focus on magnitude rather than minute detail.